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an, or the perpendicular to it. At the same time, the measures in France compared with those in Peru, give for the ellipticity of the meridian, which is less than half the former quantity. The observations of the lengths of the pendulum give the same nearly; so that this may be taken as the mean result.

The Fourth Book of the Mécanique Céleste, treats of the Tides ;—a subject on which much new light has been thrown by the investigations of Laplace.

The first satisfactory explanation which was given of the flux and reflux of the sea, was that of Newton, founded on the principle of attraction. The force of the moon acting on the terrestrial spheroid, supposing this last to be covered with water, must tend, as Newton demonstrated, to diminish the gravity of the waters toward the earth, both at the point where the moon was vertical, and at the point diametrically opposite; and this is such a ratio, that the waters would assume the figure of an oblong elliptic spheroid, with its greater axis directed to the moon. The sun must affect the great mass of the waters in a similar manner, and produce an aqueous spheroid, that at the time of new and full moon would coincide with the former, and therefore augment its effect; while at the quarters it would be at right angles to it, and in part destroy that effect.

The subject, however, was not so fully handled by Newton, but that great room appeared for improvements; and accordingly, the subject of the tides was proposed as the prize-question by the Academy of Sciences in the year 1740. This produced the three excellent dissertations of Daniel Bernoulli, Euler, and Maclaurin, which shared the prize; but shared it, we must confess, with another essay, that of Father Cavalieri, a Jesuit, who endeavoured to explain the tides by the system of vortices. It is the last time that the vortices entered the lists with the theory of gravitation.

Many excellent dissertations on the same subject have appeared since; but they are all defective in this, that they suppose the waters of the ocean in a state of equilibrium, or to be brought, by the action of gravitation, toward the earth, and toward the two other bodies just mentioned, into the figure of an aqueous spheroid, where the particles of the water, by the action of these different forces, were maintained at rest.

This, however, is by means the case: the rotation of the earth does not allow time to this spheroid ever to be accurately formed; and, long before the three attractions are able to produce their full effect, they are changed relatively to one another, and disposed to produce a different effect. Instead, therefore, of the actual formation of an aqueous spheroid, the tendency to it produces a

continual oscillation in the waters of the ocean, which are thus preserved in perpetual movement, and never can attain a state of equilibrium and of rest. To determine the nature of these oscillations, however, is a matter of extreme difficulty, and is a problem which neither Newton, nor any of the three geometers who pursued his track, was able, in the state of mechanical and mathematical science which then existed, to resolve. The best thing which they could do, was that which they actually accomplished, by inquiring into the nature of the spheroid, which, though never actually attained, was an ideal mean to which the real state of the waters made a periodical and imperfect approach. Neither the state of mechanical or mathematical science was such as could yet enable any one to determine the motions of a fluid, acted on by the three gravitations above mentioned, and having, besides, a rotatory motion. The nature of fluids was not so well known as to admit of the differential equations containing the conditions of such motions to be exhibited; and mathematical science was not so improved as to be able to integrate such equations. The first man who felt himself in possession of all the principles required to this arduous investigation, and who was bold enough to undertake a work, which, with all these resources, could not fail to involve much difficulty, was Laplace; who, in the years 1775, 1779, and 1790,

communicated to the Academy of Sciences a series of memoirs on this subject, which he has united and extended in the fourth book of the Mécanique Céleste.

Considering each particle of water as acted on by three forces, its gravitation to the earth, to the sun and to the moon, and also as impelled by the rotation of the earth, he inquires into the nature of the 'oscillations that will be excited in the fluid. He finds, that the oscillations thus arising may be divided into three classes. The first do not depend on the rotation of the earth, but only on the motion of the sun or moon in their respective orbits, and on the place of the moon's nodes. These oscillations vary periodically, but slowly; so that they do not return in the same order, till after a very long interval of time. The oscillations of the second class, depend principally on the rotation of the earth, and return in the same order, after the interval of a day nearly. The oscillations of the third class, depend on an angle that is double the angular rotation of the earth; so that they return after the interval nearly of half a day. Each of these classes of oscillations, proceeds just as if the other two had no existence; a circumstance that tends very much to simplify the investigation into their combined effect.

The oscillations of the first kind are proved to be almost entirely destroyed by the resistance which

motion of the whole sea must necessarily meet any with; and they amount nearly to the same as if the sea were reduced at every instant to an equilibrium under the attracting body.

;

The oscillations of the second class involve, in the expression of them, the rotation of the earth and they are also affected by the depth of the sea. The difference of the two tides in the same day, depends chiefly on these oscillations; and it is from thence that Laplace determines the mean depth of the sea to be about four leagues.

The oscillations of the third kind, are calculated in the same manner; and from the combination of all these circumstances, the height of the tides in different latitudes, in different situations of the sun and moon, the difference between the consecutive tides,--the difference between the time of high water and the times when the sun and moon come to the meridian,--all these circumstances are better explained in this method than they have ever been by any other theory. Laplace has instituted a very elaborate comparison between his theory and observations on the tides, made during a succession of years at Brest, a situation remarkably favourable for such observations.

1. Between the laws by which the tides diminish from their maximum at the full and change, to their minimum at the first and third quarters, and by which they increase again from the mini